Vertex slide for curves?
Straightforward way, no (currently). But possible to make it in another way.

If the cuve is as simple as a poly spline, I think you can try converting it to a mesh first, tweak it with Vertex Slide tool, then convert back to curve, but the original paramters on curve may lose then. Or:


Select the target curve point, CtrlH > Hook to New Object;
ShiftD to duplicate, then AltC2Convert the curve in to mesh object;
Tab in to the Edit Mode, select the target vertex, CtrlRMB to select the hook (a empty object) while still in Edit Mode, CtrlP > to make vertex parent. Then you can slide that vertex in the mesh to control the target curve point.


Another way is to use the handle orientation as transformation reference:


Be sure to switch Spline Type to Bezier, then set handle type to Vector;
Switch Transformation Orientation to Normal, then select the handle control point, create transformation orientation from it;
Now select the curve point, you can move it along Z axis of the new orientation.
When finished, you may want to delete the created orientation(s).